In season 10 we see that Rob & Kimberly saved money by bringing food (things like donuts, muffins, ect) from the Pit Stop and using that to eat during the leg. Thus saving some of their provided funds.
As for begging, some teams manage this exceptionally well, with several teams having no problem with getting hundreds and hundreds of dollars.
Teams will often beg for things besides money as people have bought them, books, maps, snacks, restaurant food, entry tickets, ect. In many ways the begging really has gotten way out of hand. As I hate that really from season 10 on, some teams can literally pay cabs to escort them through vast sections of the race (and yeah that isn't cheap,s o imagine how much begging that takes). Of course lately that hasn't been nearly as big of factor since they almost never drive themselves anymore (HATE that).